Pattern in Period

Date: 01/31/99 at 23:04:50
From: Michael Lau
Subject: Pattern in Period
I tried to find some periods with 1 as numerator. I found that if the
period is even, the first half of the period added to the second half
equals a series of 9's. Is there any pattern in odd periods too?
e.g.: 1/11 = 0.09090909... 0 + 9 = 9
1/7 = 0.142857142857... 142 + 857 = 999
etc.

Date: 02/02/99 at 08:50:06
From: Doctor Peterson
Subject: Re: Pattern in Period
I played with your pattern for a while to see if I could prove it to
be true. I found that it definitely isn't true for all denominators;
for 1/21, the sum of the two halves of the period is 666. But maybe
it's true for prime denominators, and also for some composites
including 14, 22, and 26.
One thing I found is that if 1/a has period 2n, and the sum of the two
halves of the period (call them x and y) is 10^n - 1, then the
equations
1 10^n x + y
--- = ----------- and x + y = 10^n - 1
a 10^(2n) - 1
can be simplified to give
a(x + 1) = 10^n + 1
so that a must be a divisor of 10^n + 1. (My equations assume the
period starts at the decimal point, restricting the problem somewhat.)
This is true, for instance, for small primes such as 7, 11, and 13,
since 1001 = 7*11*13 and for 17 (100000001 = 17*5882353) and 19
(1000000001 = 7*11*13*19*52579). I don't know enough number theory to
see easily that this is true for all primes.
But a little search found Eric Weisstein's page on Midy's theorem, ("If
the period of a repeating decimal for a/p has an even number of
digits, the sum of the two halves is a string of 9's, where p is prime
and a/p is a reduced fraction.")
http://mathworld.wolfram.com/MidysTheorem.html
This is your pattern, restricted to prime denominators, but extended
for any numerator except a multiple of the denominators. There is no
mention of any extension to odd periods.
A further search showed only that Martin Gardner discussed this theorem
in his book _Mathematical Circus_, which you might be able to find.
- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/